Pub Date : 2025-08-06DOI: 10.1016/j.aam.2025.102950
Yifeng Huang , Ruofan Jiang
We investigate the algebra and combinatorics of an analogue of the Hermite normal form that classifies finite-index submodules of . We identity both normal forms as instances of Gröbner basis theory under different monomial orders, where the Hermite normal form corresponds to the lex order, and the new normal form the hlex order. We note that the hlex normal form recovers the Smith normal form, a feature not enjoyed by the Hermite normal form. We also identify the combinatorial structure underlying the cell decomposition induced by the hlex normal form, which appears to be of independent interest. Notably, the statistics tracking the cell dimensions is compatible, in a certain way, with a collection of d “spiral shifting operators” on , which pairwise commute and collectively act freely and transitively. Using these operators, we give direct proofs of some new combinatorial identities obtained by translating the results of Solomon [26] and Petrogradsky [25] in terms of the hlex normal form.
{"title":"Lattices in Fq[[T]]d and spiral shifting operators","authors":"Yifeng Huang , Ruofan Jiang","doi":"10.1016/j.aam.2025.102950","DOIUrl":"10.1016/j.aam.2025.102950","url":null,"abstract":"<div><div>We investigate the algebra and combinatorics of an analogue of the Hermite normal form that classifies finite-index submodules of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><msup><mrow><mo>[</mo><mo>[</mo><mi>T</mi><mo>]</mo><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup></math></span>. We identity both normal forms as instances of Gröbner basis theory under different monomial orders, where the Hermite normal form corresponds to the lex order, and the new normal form the hlex order. We note that the hlex normal form recovers the Smith normal form, a feature not enjoyed by the Hermite normal form. We also identify the combinatorial structure underlying the cell decomposition induced by the hlex normal form, which appears to be of independent interest. Notably, the statistics tracking the cell dimensions is compatible, in a certain way, with a collection of <em>d</em> “spiral shifting operators” on <span><math><msup><mrow><mi>N</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, which pairwise commute and collectively act freely and transitively. Using these operators, we give direct proofs of some new combinatorial identities obtained by translating the results of Solomon <span><span>[26]</span></span> and Petrogradsky <span><span>[25]</span></span> in terms of the hlex normal form.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102950"},"PeriodicalIF":1.3,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144780617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-24DOI: 10.1016/j.aam.2025.102940
Michael Cuntz , Thorsten Holm , Peter Jørgensen
This paper studies a non-commutative generalisation of Coxeter friezes due to Berenstein and Retakh. It generalises several earlier results to this situation: A formula for frieze determinants, a T-path formula expressing the Laurent phenomenon, and results on gluing friezes together. One of our tools is a non-commutative version of the weak friezes introduced by Çanakçı and Jørgensen.
{"title":"Non-commutative friezes and their determinants, the non-commutative Laurent phenomenon for weak friezes, and frieze gluing","authors":"Michael Cuntz , Thorsten Holm , Peter Jørgensen","doi":"10.1016/j.aam.2025.102940","DOIUrl":"10.1016/j.aam.2025.102940","url":null,"abstract":"<div><div>This paper studies a non-commutative generalisation of Coxeter friezes due to Berenstein and Retakh. It generalises several earlier results to this situation: A formula for frieze determinants, a <em>T</em>-path formula expressing the Laurent phenomenon, and results on gluing friezes together. One of our tools is a non-commutative version of the weak friezes introduced by Çanakçı and Jørgensen.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102940"},"PeriodicalIF":1.0,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144696533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-23DOI: 10.1016/j.aam.2025.102938
Giuseppe Cotardo , Alberto Ravagnani , Ferdinando Zullo
We investigate the Whitney numbers of the first kind of rank-metric lattices, which are closely linked to the open problem of enumerating rank-metric codes having prescribed parameters. We apply methods from the theory of hyperovals and linear sets to compute these Whitney numbers for infinite families of rank-metric lattices. As an application of our results, we prove asymptotic estimates on the density function of certain rank-metric codes that have been conjectured in previous work.
{"title":"Whitney numbers of rank-metric lattices and code enumeration","authors":"Giuseppe Cotardo , Alberto Ravagnani , Ferdinando Zullo","doi":"10.1016/j.aam.2025.102938","DOIUrl":"10.1016/j.aam.2025.102938","url":null,"abstract":"<div><div>We investigate the Whitney numbers of the first kind of rank-metric lattices, which are closely linked to the open problem of enumerating rank-metric codes having prescribed parameters. We apply methods from the theory of hyperovals and linear sets to compute these Whitney numbers for infinite families of rank-metric lattices. As an application of our results, we prove asymptotic estimates on the density function of certain rank-metric codes that have been conjectured in previous work.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102938"},"PeriodicalIF":1.0,"publicationDate":"2025-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144686919","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-18DOI: 10.1016/j.aam.2025.102939
Fan Chung , Qizhong Lin
<div><div>For graphs <em>G</em> and <em>H</em>, we consider Ramsey numbers <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> with tight lower bounds, namely, <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>≥</mo><mo>(</mo><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mo>|</mo><mi>H</mi><mo>|</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn></math></span>, where <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> denotes the chromatic number of <em>G</em> and <span><math><mo>|</mo><mi>H</mi><mo>|</mo></math></span> denotes the number of vertices in <em>H</em>. We say <em>H</em> is <em>G</em>-good if the equality holds.</div><div>Let <span><math><mi>G</mi><mo>+</mo><mi>H</mi></math></span> be the join graph obtained from graphs <em>G</em> and <em>H</em> by adding all edges between the disjoint vertex sets of <em>G</em> and <em>H</em>. Let <em>nH</em> denote the union graph of <em>n</em> disjoint copies of <em>H</em>. We show that <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>n</mi><mi>H</mi></math></span> is <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-good if <em>n</em> is sufficiently large. In particular, the fan-graph <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>n</mi><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> is <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-good if <span><math><mi>n</mi><mo>=</mo><mi>Ω</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>, improving previous tower-type lower bounds for <em>n</em> due to Li and Rousseau (1996). Moreover, we give a stronger lower bound inequality for Ramsey number <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>F</mi><mo>)</mo></math></span> for the case of <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span>, the complete <em>p</em>-partite graph with <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. In particular, using a stability-supersaturation lemma by Fox, He and Wigderson (2023), we show that for any fixed graph <em>H</em>,<span><span><span><math><mrow><mi>r</mi><mo>(</mo><mi>G</m
{"title":"Fan-complete Ramsey numbers","authors":"Fan Chung , Qizhong Lin","doi":"10.1016/j.aam.2025.102939","DOIUrl":"10.1016/j.aam.2025.102939","url":null,"abstract":"<div><div>For graphs <em>G</em> and <em>H</em>, we consider Ramsey numbers <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> with tight lower bounds, namely, <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>≥</mo><mo>(</mo><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mo>|</mo><mi>H</mi><mo>|</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn></math></span>, where <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> denotes the chromatic number of <em>G</em> and <span><math><mo>|</mo><mi>H</mi><mo>|</mo></math></span> denotes the number of vertices in <em>H</em>. We say <em>H</em> is <em>G</em>-good if the equality holds.</div><div>Let <span><math><mi>G</mi><mo>+</mo><mi>H</mi></math></span> be the join graph obtained from graphs <em>G</em> and <em>H</em> by adding all edges between the disjoint vertex sets of <em>G</em> and <em>H</em>. Let <em>nH</em> denote the union graph of <em>n</em> disjoint copies of <em>H</em>. We show that <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>n</mi><mi>H</mi></math></span> is <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-good if <em>n</em> is sufficiently large. In particular, the fan-graph <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>n</mi><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> is <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-good if <span><math><mi>n</mi><mo>=</mo><mi>Ω</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>, improving previous tower-type lower bounds for <em>n</em> due to Li and Rousseau (1996). Moreover, we give a stronger lower bound inequality for Ramsey number <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>F</mi><mo>)</mo></math></span> for the case of <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span>, the complete <em>p</em>-partite graph with <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. In particular, using a stability-supersaturation lemma by Fox, He and Wigderson (2023), we show that for any fixed graph <em>H</em>,<span><span><span><math><mrow><mi>r</mi><mo>(</mo><mi>G</m","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102939"},"PeriodicalIF":1.0,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144654595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-15DOI: 10.1016/j.aam.2025.102936
Chunyang Dou , Bo Ning , Xing Peng
<div><div>Let <span><math><mi>H</mi></math></span> be a family of graphs. The Turán number <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> is the maximum possible number of edges in an <em>n</em>-vertex graph which does not contain any member of <span><math><mi>H</mi></math></span> as a subgraph. As a common generalization of Turán's theorem and Erdős-Gallai theorem on the Turán number of matchings, Alon and Frankl determined <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> for <span><math><mi>H</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></math></span>, where <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is a matching of size <em>k</em>. Replacing <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> by <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, Katona and Xiao obtained the Turán number of <span><math><mi>H</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></math></span> for <span><math><mi>r</mi><mo>≤</mo><mo>⌊</mo><mi>k</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></math></span> and sufficiently large <em>n</em>. In addition, they proposed a conjecture for the case where <span><math><mi>r</mi><mo>≥</mo><mo>⌊</mo><mi>k</mi><mo>/</mo><mn>2</mn><mo>⌋</mo><mo>+</mo><mn>1</mn></math></span> and <em>n</em> is sufficiently large. Motivated by the fact that the result for <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> can be deduced from the one for <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub><mo>)</mo></math></span>, we investigate the Turán number of <span><math><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub><mo>}</mo></math></span> in this paper, where <span><math><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub></math></span> denotes the set of cycles of length at least <em>k</em>. In other words, we aim to determine the maximum number of edges in graphs with clique number at most <span><math><mi>r</mi><mo>−</mo><mn>1</mn></math></span> and circumference at most <span><math><mi>k</mi><mo>−</mo><mn>1</mn></math></span>. For <span><math><mi>H</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub><mo>}</mo></math></span>, we are able to show the value of <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>
{"title":"The number of edges in graphs with bounded clique number and circumference","authors":"Chunyang Dou , Bo Ning , Xing Peng","doi":"10.1016/j.aam.2025.102936","DOIUrl":"10.1016/j.aam.2025.102936","url":null,"abstract":"<div><div>Let <span><math><mi>H</mi></math></span> be a family of graphs. The Turán number <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> is the maximum possible number of edges in an <em>n</em>-vertex graph which does not contain any member of <span><math><mi>H</mi></math></span> as a subgraph. As a common generalization of Turán's theorem and Erdős-Gallai theorem on the Turán number of matchings, Alon and Frankl determined <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> for <span><math><mi>H</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></math></span>, where <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is a matching of size <em>k</em>. Replacing <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> by <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, Katona and Xiao obtained the Turán number of <span><math><mi>H</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></math></span> for <span><math><mi>r</mi><mo>≤</mo><mo>⌊</mo><mi>k</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></math></span> and sufficiently large <em>n</em>. In addition, they proposed a conjecture for the case where <span><math><mi>r</mi><mo>≥</mo><mo>⌊</mo><mi>k</mi><mo>/</mo><mn>2</mn><mo>⌋</mo><mo>+</mo><mn>1</mn></math></span> and <em>n</em> is sufficiently large. Motivated by the fact that the result for <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> can be deduced from the one for <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub><mo>)</mo></math></span>, we investigate the Turán number of <span><math><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub><mo>}</mo></math></span> in this paper, where <span><math><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub></math></span> denotes the set of cycles of length at least <em>k</em>. In other words, we aim to determine the maximum number of edges in graphs with clique number at most <span><math><mi>r</mi><mo>−</mo><mn>1</mn></math></span> and circumference at most <span><math><mi>k</mi><mo>−</mo><mn>1</mn></math></span>. For <span><math><mi>H</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub><mo>}</mo></math></span>, we are able to show the value of <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102936"},"PeriodicalIF":1.0,"publicationDate":"2025-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144632923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-14DOI: 10.1016/j.aam.2025.102935
Jiuqiang Liu , Guihai Yu
Sperner theory is one of the most important branches in extremal set theory. It has many applications in the field of operation research, computer science, hypergraph theory and so on. The LYM property has become an important tool for studying Sperner property. In this paper, we provide a general relationship for LYM inequalities between Boolean lattices and linear lattices. As applications, we use this relationship to derive generalizations of some well-known theorems on maximum sizes of families containing no copy of certain poset or certain configuration from Boolean lattices to linear lattices, including generalizations of the well-known Kleitman theorem on families containing no s pairwise disjoint members (a non-uniform variant of the famous Erdős matching conjecture) and Johnston-Lu-Milans theorem and Polymath theorem on families containing no d-dimensional Boolean algebras.
{"title":"A relationship for LYM inequalities between Boolean lattices and linear lattices with applications","authors":"Jiuqiang Liu , Guihai Yu","doi":"10.1016/j.aam.2025.102935","DOIUrl":"10.1016/j.aam.2025.102935","url":null,"abstract":"<div><div>Sperner theory is one of the most important branches in extremal set theory. It has many applications in the field of operation research, computer science, hypergraph theory and so on. The LYM property has become an important tool for studying Sperner property. In this paper, we provide a general relationship for LYM inequalities between Boolean lattices and linear lattices. As applications, we use this relationship to derive generalizations of some well-known theorems on maximum sizes of families containing no copy of certain poset or certain configuration from Boolean lattices to linear lattices, including generalizations of the well-known Kleitman theorem on families containing no <em>s</em> pairwise disjoint members (a non-uniform variant of the famous Erdős matching conjecture) and Johnston-Lu-Milans theorem and Polymath theorem on families containing no <em>d</em>-dimensional Boolean algebras.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102935"},"PeriodicalIF":1.0,"publicationDate":"2025-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144613816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-09DOI: 10.1016/j.aam.2025.102932
V. Berthé , S. Puzynina
An infinite word generated by a substitution is rigid if all the substitutions which fix this word are powers of the same substitution. Sturmian words as well as characteristic Arnoux-Rauzy words that are generated by iterating a substitution are known to be rigid. In the present paper, we prove that all Arnoux-Rauzy words generated by iterating a substitution are rigid. The proof relies on two main ingredients: first, the fact that the primitive substitutions that fix an Arnoux-Rauzy word share a common power, and secondly, the notion of normal form of an episturmian substitution (i.e., a substitution that fixes an Arnoux-Rauzy word). The main difficulty is then of a combinatorial nature and relies on the normalization process when taking powers of episturmian substitutions: the normal form of a square is not necessarily equal to the square of the normal forms.
{"title":"On the rigidity of Arnoux-Rauzy words","authors":"V. Berthé , S. Puzynina","doi":"10.1016/j.aam.2025.102932","DOIUrl":"10.1016/j.aam.2025.102932","url":null,"abstract":"<div><div>An infinite word generated by a substitution is rigid if all the substitutions which fix this word are powers of the same substitution. Sturmian words as well as characteristic Arnoux-Rauzy words that are generated by iterating a substitution are known to be rigid. In the present paper, we prove that all Arnoux-Rauzy words generated by iterating a substitution are rigid. The proof relies on two main ingredients: first, the fact that the primitive substitutions that fix an Arnoux-Rauzy word share a common power, and secondly, the notion of normal form of an episturmian substitution (i.e., a substitution that fixes an Arnoux-Rauzy word). The main difficulty is then of a combinatorial nature and relies on the normalization process when taking powers of episturmian substitutions: the normal form of a square is not necessarily equal to the square of the normal forms.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102932"},"PeriodicalIF":1.0,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-09DOI: 10.1016/j.aam.2025.102934
S. Yakubovich
We define the generalized Dirichlet beta and Riemann zeta functions in terms of the integrals, involving powers of the hyperbolic secant and cosecant functions. The corresponding functional equations are established. Some consequences of the Ramanujan identity for zeta values at odd integers are investigated and new formulae of the Ramanujan type are obtained. These results are achieved, in particular, involving the Kontorovich-Lebedev transform and the corresponding polynomials introduced by the author.
{"title":"On the generalized Dirichlet beta and Riemann zeta functions and Ramanujan-type formulae for beta and zeta values","authors":"S. Yakubovich","doi":"10.1016/j.aam.2025.102934","DOIUrl":"10.1016/j.aam.2025.102934","url":null,"abstract":"<div><div>We define the generalized Dirichlet beta and Riemann zeta functions in terms of the integrals, involving powers of the hyperbolic secant and cosecant functions. The corresponding functional equations are established. Some consequences of the Ramanujan identity for zeta values at odd integers are investigated and new formulae of the Ramanujan type are obtained. These results are achieved, in particular, involving the Kontorovich-Lebedev transform and the corresponding polynomials introduced by the author.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102934"},"PeriodicalIF":1.0,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-08DOI: 10.1016/j.aam.2025.102933
Christine Cho, James Oxley
Let M and N be matroids such that N is the image of M under a rank-preserving weak map. Generalizing results of Lucas, we prove that, for x and y positive, if and only if or . We give a number of consequences of this result.
{"title":"Weak maps and the Tutte polynomial","authors":"Christine Cho, James Oxley","doi":"10.1016/j.aam.2025.102933","DOIUrl":"10.1016/j.aam.2025.102933","url":null,"abstract":"<div><div>Let <em>M</em> and <em>N</em> be matroids such that <em>N</em> is the image of <em>M</em> under a rank-preserving weak map. Generalizing results of Lucas, we prove that, for <em>x</em> and <em>y</em> positive, <span><math><mi>T</mi><mo>(</mo><mi>M</mi><mo>;</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>≥</mo><mi>T</mi><mo>(</mo><mi>N</mi><mo>;</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> if and only if <span><math><mi>x</mi><mo>+</mo><mi>y</mi><mo>≥</mo><mi>x</mi><mi>y</mi></math></span> or <span><math><mi>M</mi><mo>≅</mo><mi>N</mi></math></span>. We give a number of consequences of this result.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102933"},"PeriodicalIF":1.0,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144572076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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